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  "body": "\n## Definition\n\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be a [[functor|28NQ0KRHD3]] between [[categories|A2DE66937D]] and let $X$ be an object of $\\mathcal{D}$.\nA *universal morphism from $X$ to $F$* is a pair $(A, u)$, with $A$ in $\\mathcal{C}$ and $u \\colon X \\to F(A)$ in $\\mathcal{D}$, with this property.\nFor every object $A'$ of $\\mathcal{C}$ and every morphism $f \\colon X \\to F(A')$, there is exactly one $g \\colon A \\to A'$ with $F(g) \\circ u = f$.\nDually, a *universal morphism from $F$ to $X$* is a pair $(A, v)$ with $v \\colon F(A) \\to X$, with the following property.\nFor every $A'$ and every $f \\colon F(A') \\to X$, there is exactly one $g \\colon A' \\to A$ with $v \\circ F(g) = f$.\nAn object or morphism is said to have a *universal property* when it is part of such a pair.\n\n## Remarks\n\nEquivalently, $(A, u)$ is an initial object of the comma category $(X \\downarrow F)$, whose objects are pairs $(A', f \\colon X \\to F(A'))$; in the dual case it is a terminal object of $(F \\downarrow X)$.\nSince [[initial objects|EK85VSSRMR]] are unique up to unique isomorphism, so are universal morphisms.\n\nExample: let $U \\colon \\mathbf{Grp} \\to \\mathbf{Set}$ send a group to its underlying set, and let $S$ be a set.\nThe [[free group|0KTW26C80J]] $F(S)$ with the inclusion $u \\colon S \\to U(F(S))$ of generators is a universal morphism from $S$ to $U$.\nIndeed, every function $f \\colon S \\to U(H)$ to a group $H$ extends to exactly one [[group homomorphism|0JYTG5K9Y4]] $g \\colon F(S) \\to H$ with $g \\circ u = f$.\n\nNon-example: take $S = \\{s\\}$ and the pair $(\\Z \\times \\Z, \\ s \\mapsto (1, 0))$.\nThe function $s \\mapsto 1 \\in \\Z$ extends to the homomorphisms $(m, n) \\mapsto m$ and $(m, n) \\mapsto m + n$, so uniqueness fails and this pair is not universal.\n",
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